Smith theory and cyclic base change functoriality
Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\math...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Pi |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S205050862300032X/type/journal_article |
| _version_ | 1797354988132892672 |
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| author | Tony Feng |
| author_facet | Tony Feng |
| author_sort | Tony Feng |
| collection | DOAJ |
| description | Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For
$\mathbf {Z}/p\mathbf {Z}$
-extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For
$\mathbf {Z}/p\mathbf {Z}$
-extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along
$\mathbf {Z}/p\mathbf {Z}$
-extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh. |
| first_indexed | 2024-03-08T13:57:45Z |
| format | Article |
| id | doaj.art-176ebc1ec576431d911c1baba832f9df |
| institution | Directory Open Access Journal |
| issn | 2050-5086 |
| language | English |
| last_indexed | 2024-03-08T13:57:45Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj.art-176ebc1ec576431d911c1baba832f9df2024-01-15T07:57:57ZengCambridge University PressForum of Mathematics, Pi2050-50862024-01-011210.1017/fmp.2023.32Smith theory and cyclic base change functorialityTony Feng0https://orcid.org/0000-0002-5150-5733University of California, Berkeley, 94720, USA;Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $\mathbf {Z}/p\mathbf {Z}$ -extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along $\mathbf {Z}/p\mathbf {Z}$ -extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh.https://www.cambridge.org/core/product/identifier/S205050862300032X/type/journal_article11S3711F80 |
| spellingShingle | Tony Feng Smith theory and cyclic base change functoriality Forum of Mathematics, Pi 11S37 11F80 |
| title | Smith theory and cyclic base change functoriality |
| title_full | Smith theory and cyclic base change functoriality |
| title_fullStr | Smith theory and cyclic base change functoriality |
| title_full_unstemmed | Smith theory and cyclic base change functoriality |
| title_short | Smith theory and cyclic base change functoriality |
| title_sort | smith theory and cyclic base change functoriality |
| topic | 11S37 11F80 |
| url | https://www.cambridge.org/core/product/identifier/S205050862300032X/type/journal_article |
| work_keys_str_mv | AT tonyfeng smiththeoryandcyclicbasechangefunctoriality |